An adaptive wavelet method for solving high-dimensional elliptic PDEs

Constructive Approximation

Volumn 30, Issue 3, 2009, Pages 423-455

  Dijkema, Tammo Jan ^a   Schwab, Christoph ^b   Stevenson, Rob ^c  

^a UTRECHT UNIVERSITY   (Netherlands)

^b ETH ZURICH   (Switzerland)

^c UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

Adaptive wavelet methods; Best N term approximations; Matrix compression; Optimal computational complexity; Sparse grids; Tensor product approximation

Indexed keywords

EID: 70449528428     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-009-9064-0     Document Type: Article

References (30)

1. Barinka, A.: Fast evaluation tools for adaptive wavelet schemes. PhD thesis, RTWH Aachen, March 2005.
2. Bungartz, H. J., Griebel, M.: Sparse grids. Acta Numer. 13, 147-269 (2004).
3. Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam (2003).
4. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations-convergence rates. Math. Comput. 70, 27-75 (2001).
5. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods II-Beyond the elliptic case. Found. Comput. Math. 2(3), 203-245 (2002).
6. Dahmen, W., Harbrecht, H., Schneider, R.: Compression techniques for boundary integral equations-optimal complexity estimates. SIAM J. Numer. Anal. 43(6), 2251-2271 (2006).
7. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909-996 (1988).
8. Dauge, M.: Elliptic Boundary Value Problems on Corner Domain, Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988).
9. DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51-150 (1998).
10. Dijkema, T. J.: Adaptive tensor product wavelet methods for solving PDEs. PhD thesis, Utrecht University (June 2009).
11. Donovan, G. C., Geronimo, J. S., Hardin, D. P.: Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets. SIAM J. Math. Anal. 27(6), 1791-1815 (1996).
12. Donovan, G. C., Geronimo, J. S., Hardin, D. P.: Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30(5), 1029-1056 (1999).
13. Feynman, A. R., Hibbs, A. R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965).
14. Freidlin, M.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, vol. 109. Princeton University Press, Princeton (1985).
15. Gantumur, T., Stevenson, R. P.: Computation of differential operators in wavelet coordinates. Math. Comput. 75, 697-709 (2006).
16. Gantumur, T., Harbrecht, H., Stevenson, R. P.: An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput. 76, 615-629 (2007).
17. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems. Computing 74(2), 131-157 (2005).
18. Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72(3-4), 247-265 (2004).
19. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulation. J. Comput. Phys. 73, 325-348 (1987).
20. Griebel, M., Knapek, S.: Optimized tensor-product approximation spaces. Constr. Approx. 16(4), 525-540 (2000).
21. Griebel, M., Oswald, P.: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4(1-2), 171-206 (1995).
22. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Applied Mathematical Sciences, vol. 95. Springer, New York (1994).
23. Hoang, V. H., Schwab, Ch.: High-dimensional finite elements for elliptic problems with multiple scales. SIAM J. Multiscale Model. Simul. 3(1), 168-194 (2005).
24. Metselaar, A.: Handling wavelet expansions in numerical methods. PhD thesis, University of Twente (2002).
25. Nitsche, P.-A.: Sparse approximation of singularity functions. Constr. Approx. 21(1), 63-81 (2005).
26. Nitsche, P.-A.: Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx. 24(1), 49-70 (2006).
27. Schwab, Ch., Stevenson, R. P.: Adaptive wavelet algorithms for elliptic PDEs on product domains. Math. Comput. 77, 71-92 (2008).
28. Todor, R. A., Schwab, Ch.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27(2), 232-261 (2007).
29. Yserentant, H.: Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101(2), 381-389 (2005).
30. Zenger, Ch.: Sparse grids. In: Parallel Algorithms for Partial Differential Equations, Kiel, 1990. Notes Numer. Fluid Mech., vol. 31, pp. 241-251. Vieweg, Braunschweig (1991).